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A055231 Powerfree part of n: product of primes that divide n only once. 41
1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 1, 73, 74, 3, 19, 77, 78, 79, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The previous name was: Write n = K^2*F where F is squarefree and F = g*f where g = gcd(K,F) and f = F/g; then a(n) = f(n) = F(n)/g(n). Thus gcd(K^2,f) = 1.

Differs from A007913; they coincide if and only if g(n) = 1.

a(n) is the powerfree part of n; i.e., if n=Product(pi^ei) over all i [prime factorization) then a(n)=Product(pi^ei) over those i with ei=1; if n=b*c^2*d^3 then a(n) is minimum possible value of b. - Henry Bottomley, Sep 01 2000

Also denominator of n/rad(n)^2, where rad is the squarefree kernel of n (A007947), numerator: A062378. - Reinhard Zumkeller, Dec 10 2002

Largest unitary squarefree number dividing n (the unitary squarefree kernel of n). - Steven Finch, Mar 01 2004

From Bernard Schott, Mar 24 2020: (Start)

a(n) = 1 iff n is a perfect power (A001597),

a(n) = n iff n is a squarefree number (A005117), and

(a(n) <> 1 and a(n) <>n) iff n is in A059404. (End)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]

Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

FORMULA

a(n) = n/A057521(n).

Multiplicative with a(p) = p and a(p^e) = 1 for e > 1. - Vladeta Jovovic, Nov 01 2001

Dirichlet g.f.: zeta(s)*Product_{primes p} (1 + p^(1-s) - p^(-s) - p^(1-2s) + p^(-2s)). - R. J. Mathar, Dec 21 2011

a(n) = A007947(n)/A071773(n). - observed by Velin Yanev, Aug 27 2017, confirmed by Antti Karttunen, Nov 28 2017

a(1) = 1; for n > 1, a(n) = A020639(n)^A063524(A067029(n)) * a(A028234(n)). - Antti Karttunen, Nov 28 2017

a(n*m) = a(n)*a(m)/(gcd(n,a(m))*gcd(m,a(n))) for all n and m > 0 (conjectured). - Velin Yanev, Feb 06 2019. [This follows easily from the comment of Vladeta Jovovic. - N. J. A. Sloane, Mar 14 2019]

From Vaclav Kotesovec, Dec 19 2019: (Start)

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 - p^(1-3*s) + p^(2-3*s) - p^(2-2*s) + p^(-2*s) - p^(-s)).

Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = Product_{primes p} (1 - 2/p^2 + 2/p^4 - 1/p^5) = 0.394913518073109872954607634745304266741971541072... (End)

EXAMPLE

If n = 15!, A008833(15!) = 30240*30240, A007913(15!) = 1430, g(15!) = 10, a(15!) = A007913(15!) = 143 and GCD[30240,143] = 1. 15! = (30240*30240)*1430 = (30240^2)*10*143 = K*K*F = (K^2)*g*f.

MAPLE

A055231 := proc(n)

    a := 1 ;

    if n > 1 then

        for f in ifactors(n)[2] do

            if op(2, f) = 1 then

                a := a*op(1, f) ;

            end if;

        end do:

    end if;

    a ;

end proc: # R. J. Mathar, Dec 23 2011

MATHEMATICA

rad[n_] := Times @@ First /@ FactorInteger[n]; a[n_] := Denominator[n/rad[n]^2]; Table[a[n], {n, 1, 80}] (* Jean-Fran├žois Alcover, Jun 20 2013, after Reinhard Zumkeller *)

f[p_, e_] := If[e==1, p, 1]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

PROG

(PARI) A055231(n)={

   local(a=1);

   f=factor(n) ;

   for(i=1, matsize(f)[1],

         if( f[i, 2] ==1, a *=  f[i, 1]

         )

   ) ;

   a ;

} /* R. J. Mathar, Mar 12 2012 */

(PARI) a(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); } \\ Michel Marcus, Aug 27 2017

(Scheme, with memoization-macro definec) (definec (A055231 n) (if (= 1 n) 1 (* (if (= 1 (A067029 n)) (A020639 n) 1) (A055231 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

CROSSREFS

a(n) = A007913(n)/gcd(A008833(n!), A007913(n!)).

Cf. A008833, A007913, A007947, A000188, A057521, A055773 (computed for n!), A056169 (number of prime divisors), A056671 (number of divisors), A092261 (sum of divisors of the n-th term).

Cf. A005117 (subsequence).

Sequence in context: A130508 A182938 A329445 * A304328 A304339 A160400

Adjacent sequences:  A055228 A055229 A055230 * A055232 A055233 A055234

KEYWORD

nonn,mult

AUTHOR

Labos Elemer, Jun 21 2000

EXTENSIONS

Name replaced with a simpler description (based on Henry Bottomley's comment) by Antti Karttunen, Nov 28 2017

STATUS

approved

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Last modified September 21 13:20 EDT 2020. Contains 337272 sequences. (Running on oeis4.)